Calculus Examples

$f (x) = \frac{\cos (t)}{t^{5}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = \cos (t)$ and $g (t) = t^{5}$ .

$\frac{t^{5} \frac{d}{d t} [\cos (t)] - \cos (t) \frac{d}{d t} [t^{5}]}{{(t^{5})}^{2}}$

Step 2

The derivative of $\cos (t)$ with respect to $t$ is $- \sin (t)$ .

$\frac{t^{5} (- \sin (t)) - \cos (t) \frac{d}{d t} [t^{5}]}{{(t^{5})}^{2}}$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 5$ .

$\frac{t^{5} (- \sin (t)) - \cos (t) (5 t^{4})}{{(t^{5})}^{2}}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Simplify each term.

Tap for more steps...

Step 4.1.1.1

Rewrite using the commutative property of multiplication.

$\frac{- t^{5} \sin (t) - \cos (t) (5 t^{4})}{{(t^{5})}^{2}}$

Step 4.1.1.2

Multiply $5$ by $- 1$ .

$\frac{- t^{5} \sin (t) - 5 \cos (t) t^{4}}{{(t^{5})}^{2}}$

Step 4.1.2

Reorder factors in $- t^{5} \sin (t) - 5 \cos (t) t^{4}$ .

$\frac{- t^{5} \sin (t) - 5 t^{4} \cos (t)}{{(t^{5})}^{2}}$

Step 4.2

Combine terms.

Tap for more steps...

Step 4.2.1

Multiply the exponents in ${(t^{5})}^{2}$ .

Tap for more steps...

Step 4.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{- t^{5} \sin (t) - 5 t^{4} \cos (t)}{t^{5 \cdot 2}}$

Step 4.2.1.2

Multiply $5$ by $2$ .

$\frac{- t^{5} \sin (t) - 5 t^{4} \cos (t)}{t^{10}}$

Step 4.2.2

Factor $t^{4}$ out of $- t^{5} \sin (t) - 5 t^{4} \cos (t)$ .

Tap for more steps...

Step 4.2.2.1

Factor $t^{4}$ out of $- t^{5} \sin (t)$ .

$\frac{t^{4} (- t \sin (t)) - 5 t^{4} \cos (t)}{t^{10}}$

Step 4.2.2.2

Factor $t^{4}$ out of $- 5 t^{4} \cos (t)$ .

$\frac{t^{4} (- t \sin (t)) + t^{4} (- 5 \cos (t))}{t^{10}}$

Step 4.2.2.3

Factor $t^{4}$ out of $t^{4} (- t \sin (t)) + t^{4} (- 5 \cos (t))$ .

$\frac{t^{4} (- t \sin (t) - 5 \cos (t))}{t^{10}}$

Step 4.2.3

Cancel the common factors.

Tap for more steps...

Step 4.2.3.1

Factor $t^{4}$ out of $t^{10}$ .

$\frac{t^{4} (- t \sin (t) - 5 \cos (t))}{t^{4} t^{6}}$

Step 4.2.3.2

Cancel the common factor.

$\frac{t^{4} (- t \sin (t) - 5 \cos (t))}{t^{4} t^{6}}$

Step 4.2.3.3

Rewrite the expression.

$\frac{- t \sin (t) - 5 \cos (t)}{t^{6}}$

Step 4.3

Factor $- 1$ out of $- t \sin (t)$ .

$\frac{- (t \sin (t)) - 5 \cos (t)}{t^{6}}$

Step 4.4

Factor $- 1$ out of $- 5 \cos (t)$ .

$\frac{- (t \sin (t)) - (5 \cos (t))}{t^{6}}$

Step 4.5

Factor $- 1$ out of $- (t \sin (t)) - (5 \cos (t))$ .

$\frac{- (t \sin (t) + 5 \cos (t))}{t^{6}}$

Step 4.6

Rewrite $- (t \sin (t) + 5 \cos (t))$ as $- 1 (t \sin (t) + 5 \cos (t))$ .

$\frac{- 1 (t \sin (t) + 5 \cos (t))}{t^{6}}$

Step 4.7

Move the negative in front of the fraction.

$- \frac{t \sin (t) + 5 \cos (t)}{t^{6}}$